Pairwise Disjoint Balls on a Manifold

Question

I am wondering if it is always possible to find disjoint sets on any manifold such that these sets are balls when mapped to their locally Euclidean space $such$ $that$ there are an infinite number of such sets.

The result is, for example, obvious when the manifold is itself Euclidean; not sure if this is true in general.

Answer 1

You could always look at an open $\Bbb R^n$-homeomorphic subset of your manifold and say "this is basically Euclidian, so I can find an infinitude of balls within this subset." Then you're done.

Also, note that for any open subset of a manifold homeomorphic to Euclidian space, there is no unique such homeomorphism, so a set that looks like a ball under one homeomorphism can look like a cube under another.